93 lines
3.4 KiB
Python
93 lines
3.4 KiB
Python
import math
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# 求差分表
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def GetDyList(list_y):
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result = []
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result.append(list_y)
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for i in range(len(list_y)-1,0,-1):
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tmp = []
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for j in range(i):
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tmp.append(result[len(list_y)-1-i][j+1] - result[len(list_y)-1-i][j])
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result.append(tmp)
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return result
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# 牛顿前插余项计算
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def NewtonForwardRegression(x,n,list_x,FxDiff_n):
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h = list_x[1] - list_x[0]
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t = (x - list_x[0]) / h
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ks = max([abs(FxDiff_n(list_x[i],n)) for i in range(n)])
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result = 1
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for i in range(n):
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result *= (t-i)*h/(i+1)
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return result * ks
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# 牛顿前插
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def NewtonForwardInterpolation(x,n,list_x,list_y,FxDiff_n):
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list_dyk = GetDyList(list_y)
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print("\n差分表")
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print("--------------------------------------------------")
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for i in range(len(list_dyk)):
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print(list_dyk[i])
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print("--------------------------------------------------")
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h = list_x[1] - list_x[0]
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t = (x - list_x[0]) / h
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result = list_y[0]
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mul = 1
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result = list_y[0]
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for i in range(1, n):
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mul *= ((t-i+1)/i)
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result += list_dyk[i][0] * mul
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r = abs(NewtonForwardRegression(x,n,list_x,FxDiff_n))
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return (result,r)
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# 求差商表
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def GetDQyList(list_x,list_y):
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result = []
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result.append(list_y)
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for i in range(len(list_y)-1,0,-1):
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tmp = []
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for j in range(i):
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tmp.append((result[len(list_y)-1-i][j+1] - result[len(list_y)-1-i][j])/(list_x[j+len(list_y)-i] - list_x[j]))
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result.append(tmp)
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return result
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# 牛顿基本插值
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def NewtonBaseInterpolation(x,n,list_x,list_y):
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list_dqy = GetDQyList(list_x,list_y)
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print("\n差商表")
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print("--------------------------------------------------")
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for i in range(len(list_dqy)):
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print(list_dqy[i])
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print("--------------------------------------------------")
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result = list_dqy[0][0]
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mul = 1
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for i in range(1,n):
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mul *= (x - list_x[i-1])
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result += list_dqy[i][0] * mul
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return result
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# 定义原函数和其导函数计算结果，用于计算牛顿前插的截断误差，如果不需要则不用管
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def FxDiff_n1(x,n):
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result = 0
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if n == 0:
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# 下面改成原函数 ############################################################
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result = math.exp(x)
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else:
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# 下面改成n阶导数 ##############################################################################
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result = math.exp(x)
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return result
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if __name__ == '__main__':
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##############################################################################################################
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list_x = [0.0,0.2,0.4,0.6,0.8] # 已给出的x数值，与y数值对应
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list_y = [1.0,1.2214,1.4918,1.8221,2.2255] # 已给出的y数值，与x数值对应
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x_to_predict = 0.12 # 要预测的x值
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# 记得改下面的点数(第二个参数)，n是点数，阶数为n-1 ################################
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print("三点牛顿前插结果为%f, 截断误差%f" % NewtonForwardInterpolation(x_to_predict, 3, list_x, list_y,FxDiff_n1))
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print("四点牛顿前插结果为%f, 截断误差%f" % NewtonForwardInterpolation(x_to_predict, 4, list_x, list_y,FxDiff_n1))
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print("三点牛顿基本插值结果为%f" % NewtonBaseInterpolation(x_to_predict, 3, list_x, list_y))
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print("四点牛顿基本插值结果为%f" % NewtonBaseInterpolation(x_to_predict, 4, list_x, list_y)) |